Source-linked AI summary
Quantum theory cannot consistently describe the use of itself
Daniela Frauchiger, Renato Renner
TL;DR
The paper investigates whether quantum theory can have universal validity when used to describe complex systems containing agents who themselves use quantum theory. It analyzes a self-referential Gedankenexperiment and finds that the agents derive contradictory claims, indicating that straightforward extrapolation to complex systems fails.
Problem
Quantum theory is highly accurate for fundamental processes, but whether it has universal validity beyond mostly microscopic experimentally tested domains remains open.
Method
The paper proposes a Gedankenexperiment in which multiple agents use quantum theory to reason about other agents who also use it.
Results
The agents derive contradictory statements within quantum theory, including conclusions that assign opposite outcomes with certainty.
Takeaways & Limitations
The result indicates that quantum theory cannot be extrapolated to complex systems in a straightforward manner.
Takeaways & Limitations
The proposed experiment is not claimed to be technologically feasible in its presented form and is intended to scrutinize theoretical consistency rather than probe nature.
Abstract
from arXiv · showhide
Quantum theory provides an extremely accurate description of fundamental processes in physics. It thus seems likely that the theory is applicable beyond the, mostly microscopic, domain in which it has been tested experimentally. Here we propose a Gedankenexperiment to investigate the question whether quantum theory can, in principle, have universal validity. The idea is that, if the answer was yes, it must be possible to employ quantum theory to model complex systems that include agents who are themselves using quantum theory. Analysing the experiment under this presumption, we find that one agent, upon observing a particular measurement outcome, must conclude that another agent has predicted the opposite outcome with certainty. The agents' conclusions, although all derived within quantum theory, are thus inconsistent. This indicates that quantum theory cannot be extrapolated to complex systems, at least not in a straightforward manner.
1 Introduction
The paper asks whether quantum theory can remain universally valid when applied to complex systems containing agents who themselves use the theory. Extending Wigner’s setup, it argues that self-referential quantum reasoning produces contradictory conclusions under three natural assumptions.
- Motivation: Quantum theory is experimentally tested mostly in microscopic domains but is also used to describe larger-scale phenomena, motivating questions about its universal validity.Examples include the cosmic microwave background and black-hole radiation.
- Wigner’s friend: Wigner’s friend setup gives the friend a pure state for the measured spin, while an outside observer models the isolated laboratory as a quantum system in superposition.The outside observer lacks access to the measurement outcome and can therefore assign a superposition to the laboratory.
- Earlier arguments: The differing state assignments need not contradict one another because they reflect the agents’ different information about the measurement outcome.The friend has observed z, whereas the outside observer has not; restricting the laboratory state to the spin gives a maximally mixed state rather than the friend’s pure state.
- Present proposal: The proposed Gedankenexperiment extends Wigner’s setup to agents reasoning about other agents who also use quantum theory.Its main finding is formulated as a no-go theorem involving assumptions (Q), (C), and (S).
- Present proposal: Assumptions (Q), (C), and (S) cannot all hold: universal quantum certainty, consistency of predictions, and one definite outcome for each measurement conflict.The theorem does not identify which assumption must be rejected, but it creates a conflict for interpretations applying quantum theory to the experiment.
2 Results
The Gedankenexperiment combines isolated labs, quantum measurements, and nested reasoning among four agents. Under assumptions (Q), (C), and (S), their quantum-theoretic conclusions become contradictory, yielding a no-go theorem.
- The Gedankenexperiment: The experiment makes information about F’s spin outcome available externally through a random preparation value r while preserving the isolation of F’s lab.This extends Wigner’s setup by allowing an outside agent to know information correlated with z.
- The Gedankenexperiment: The protocol uses four agents in separate labs: ¯F randomizes and prepares S, F measures its vertical polarization, and ¯W and W measure the two labs.The rounds repeat until both outside agents announce ok.
- Scope: The argument is a thought experiment rather than a claim of technological feasibility, and it relies on concepts such as agents and time.The authors note that theories with non-standard understandings of these concepts might avoid the conclusion, although no concrete examples are known.
- Analysis: Assumption (Q) lets each agent use the Born rule to predict measurement outcomes with certainty, including measurements on systems containing other agents.The analysis applies this rule to the agents’ differing information and nested conclusions.
- Analysis: In the final round, W infers with certainty that w = fail from ¯W’s announced information, yet observes w = ok, producing a contradiction under Assumption (S).Assumption (S) disallows simultaneously assigning multiple values to w.
- No-go theorem: The no-go theorem states that any theory satisfying (Q), (C), and (S) yields contradictory statements in this Gedankenexperiment.The result is used to classify interpretations according to which assumption they reject.
- Interpretations: For subsystem-based hidden-variable theories, the analysis concludes that even assigning hidden-variable values consistent with the agents’ conclusions is impossible in general.The discussion distinguishes these theories from applying the laws to the universe as a whole.
3 Discussion
The Gedankenexperiment shows that agents reasoning about one another within quantum theory can reach contradictory statements, without counterfactual choices. The discussion compares this result with interpretations, earlier no-go theorems, and a possible computer-based test.
- Implications: Nested reasoning about agents using quantum theory can produce contradictory statements even when the agents’ conclusions concern classical cases.The setup restricts the agents’ conclusions to supposedly unproblematic classical information, yet contradictions remain.
- Implications: The contradiction can be framed as a dispute over incompatible gambling claims about the same experimental outcomes.The proposed game leaves the judge to consider arguments implying different values for the random variable r.
- Relation to earlier results: The no-go theorem complements earlier results by using assumptions resembling (Q) and (S) without adding assumptions about reality, locality, or freedom of choice.The paper presents its theorem as differing in the assumptions used to derive inconsistency.
- Relation to earlier results: The argument avoids counterfactual reasoning: agents make no choices, and their statements concern no values that are unavailable when stated.This distinguishes the proposal from earlier no-go arguments involving unrealized alternatives or delayed choices.
- Possible experimental test: A modified protocol could replace the agents with computers whose information processing and isolation requirements might be realized using quantum computers.The paper suggests experimentally testing statements in Table 3 by reading out the computers’ outputs during the procedure.
A.1 Information-theoretic description
The circuit diagram represents the Gedankenexperiment in information-theoretic terms: every agent knows the overall evolution, but each accesses different data.
- A.1 Information-theoretic description: The circuit diagram encodes the experiment’s information structure through wires representing the different data available to the agents.All agents have full information about the overall circuit evolution, while their accessible data differ.
A.2 Derivation of statement W0:00 using Assumption (Q)
The event (w̄, w) = (ok, ok) has positive probability, so it occurs after finitely many rounds; this supports statement W0:00 under Assumption (Q).
- A.2 Derivation of statement W0:00 using Assumption (Q): The event (w̄, w) = (ok, ok) occurs after finitely many rounds, as stated by W0:00.The analysis first establishes finite-round occurrence and then shows it follows from Assumption (Q), not only the full Born rule.
- A.2 Derivation of statement W0:00 using Assumption (Q): The derivation represents the round-n event with a Heisenberg operator and inserts that measurement operator into Assumption (Q).The operator construction and substitution connect the event calculation to the certainty statement W0:00.
- A.2 Derivation of statement W0:00 using Assumption (Q): 1/12 > 0: the event (w̄, w) = (ok, ok) has positive probability in each relevant round.The calculation identifies the event probability used in deriving W0:00.
A.3 Analysis within Bohmian mechanics
Within Bohmian mechanics, applying the equations to the agents’ local systems reproduces the reasoning required by (Q), while (S) then forces a violation of consistency (C). Applying the theory to the entire universe instead avoids that violation by requiring an outside perspective, but necessarily violates (Q).
- Analysis within Bohmian mechanics: Applying Bohmian equations directly to the relevant systems around the agents yields the reasoning prescribed by (Q).This treats the agents’ surrounding systems rather than the universe as a whole.
- Analysis within Bohmian mechanics: Because Bohmian mechanics also satisfies (S), the no-go theorem implies a violation of (C), so the agents’ conclusions contradict each other.The result concerns the agents’ self-referential conclusions within the experiment.
- Analysis within Bohmian mechanics: Applying Bohmian mechanics to the entire universe makes the agents model themselves from an outside perspective, keeping their views aligned and reasoning under (C) unproblematic.This is the perspective required by the directive to apply Bohmian mechanics universally.
- Analysis within Bohmian mechanics: The no-go theorem then requires (Q) to fail, confirmed by an explicit calculation showing that statement ¯Fn:02 does not hold.The result departs from standard quantum mechanics, where the order of the separate measurements is irrelevant.
- Analysis within Bohmian mechanics: If W measures before ¯W, Bohmian mechanics instead invalidates ¯Wn:22 while ¯Fn:02 holds, making measurement order relevant.This time-order dependence is identified as a departure from standard quantum mechanics.
- Analysis within Bohmian mechanics: The analysis leaves open when Bohmian mechanics still endorses the Born rule, despite a proposed memory-based criterion failing in the relevant case.When ¯w = ok, r and ¯F’s prediction for w remain retrievable when w is measured.
A.4 Analysis within the CH formalism
The CH formalism represents measurement claims as histories belonging to consistency-constrained frameworks. It can support incompatible inferences when histories from different frameworks are compared, thereby satisfying (Q) and (S) but violating (C).
- Analysis within the CH formalism: In CH, measurement-outcome statements are histories that must belong to a framework satisfying consistency conditions.A possible history in this experiment is denoted h1.
- Analysis within the CH formalism: History h1 is verified by constructing a framework containing it together with additional histories.The passage describes this framework construction as straightforward.
- Analysis within the CH formalism: The CH formalism contains the Born rule as a special case and therefore fulfils (Q); because it also satisfies (S), the no-go theorem implies a violation of (C).The violation is illustrated using a shortened history h′1 that omits z and ¯w.
- Analysis within the CH formalism: CH accounts for incompatible conclusions by restricting logical reasoning to histories within a single framework.Histories h1 and h′1 cannot be combined for unrestricted inference.
- Analysis within the CH formalism: The framework containing h′1 makes the gambler’s reasoning correct because Pr[h′1] = 0, while the framework containing h1 assigns nonzero probabilities to alternative histories.In the latter framework, all nonzero-probability histories agreeing with ¯w = ok also assert z = + 1/2.
- Analysis within the CH formalism: Within one framework, ¯w = ok implies z = + 1/2 and r = tails, but the framework does not include a history about r alone.Consequently, it disallows the implication from r = tails and z = + 1/2 to r = tails.
A.5 Analysis within QBism
QBism treats quantum states and measurement outcomes as personal to an agent. In the Gedankenexperiment, allowing agents to draw certain nested implications produces contradictory statements, so consistency requires restricting such implications or replacing (C) with a weaker rule.
- Analysis within QBism: QBism regards quantum states as representations of an agent’s personal knowledge or beliefs about future measurement outcomes, which are also personal to that agent.This subjectivistic interpretation frames the agents’ notebook statements as personal conclusions.
- Analysis within QBism: Agent ¯F may record that observing r = tails makes her certain that W will announce w = fail at the round’s end.The certainty is expressed as a degree of belief, potentially equivalent to betting an arbitrarily large amount.
- Analysis within QBism: Agent F may reason about ¯F’s notebook and conclude that it records r = tails, then reason further about ¯F’s certainty regarding W’s announcement.These are nested conclusions about another agent’s observations and beliefs.
- Analysis within QBism: Permitting implications such as F1:13 =⇒ F1:14 is equivalent to assuming (C), but QBism’s satisfaction of (Q) and (S) then yields contradictory statements.The paper therefore disallows such implications in this multi-agent scenario.
- Analysis within QBism: QBism’s consistent use in this scenario requires disallowing implications of the type F1:13 =⇒ F1:14.The possibility of replacing (C) with a weaker consistency rule remains an open question under investigation.